Alphabeta Math
CorollaryStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-07-31
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Every convergent real series is Cesaro summable and Abel summable to its ordinary sum

Statement

If ∑n≥0an\sum_{n\ge0}a_n converges ordinarily to ss, then it is both Cesaro summable and Abel summable to ss.

Facts & Assumptions

Given: Partial sums Sn→sS_n\to s.

[L1]

The Cesaro means of a convergent sequence converge to the same limit (If xk→Lx_k \to L then σn→L\sigma_n \to L: convergence implies (C,1)(C,1)-summability to the same value).

[L2]

Abel's limit theorem sends an ordinarily convergent series to its ordinary sum (Abel's limit theorem: if a real series converges to ss, then its power series tends to ss as x↑1x\uparrow1).

Proof

technique · direct
1.1

Apply [L1] to (Sn)(S_n) to obtain σn→s\sigma_n\to s, which is Cesaro summability.

givenL1
2.1

Apply [L2] to the original series to obtain Abel summability to ss.

givenL2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 71 results over 14 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources